Pick over the strawberries, removing the hulls and discarding any blemished berries.
Wash the strawberries briefly in a large sieve or colander set under cold running water.
Then spread them on paper towels to drain and pat them completely dry with fresh towels.
Select 2 cups of the most attractive berries and set them aside.
Drop the remaining berries into a bowl, add 1/2 cup of the sugar, and mash the berries to a thick puree by beating them against the sides of the bowl with the back of a spoon.
Cover the bowl with foil or plastic wrap and let the pureed strawberries steep at room temperature until ready to serve.
Preheat the oven to 350°.
With a pastry brush, spread the softened butter evenly over the bottom and sides of two 9-inch layer-cake pans.
Add 1 tablespoon of the flour to each pan and tip it from side to side to spread the flour evenly.
Invert the pans and rap their bottoms sharply to remove the excess flour.
Combine the remaining cup of flour and the salt and sift them together onto a plate or a sheet of wax paper.
Set aside.
With a wire whisk or a rotary or electric beater, beat the egg whites until they are stiff enough to stand in unwavering peaks on the beater when it is lifted from the bowl.
In another bowl and with the same beater, beat the egg yolks, 1/2 cup of the sugar, and 1 teaspoon of the vanilla together until they are thick and lemon-colored.
Sprinkle the sifted flour and salt over the egg whites, pour the egg-yolk mixture over them and, with a rubber spatula, fold together lightly but thoroughly, using an over-and-under cutting motion rather than a stirring motion.
Pour the batter into the buttered-and-floured pans, dividing it equally among them.
Spread the batter and smooth the tops with the spatula.
Bake in the middle of the oven for about 20 minutes, or until the cakes begin to shrink away from the sides of the pans and a cake tester or toothpick inserted in the centers comes out clean.
Let the cakes cool in the pans for 4 or 5 minutes, then turn them out onto wire racks to cool completely to room temperature.
About half an hour before serving the cake, pour the cream into a large chilled bowl and whip it with a wire whisk or a rotary or electric beater.
As soon as the cream becomes frothy, add the remaining 2 tablespoons of sugar and 1 teaspoon of vanilla.
Continue to beat until the cream is stiff enough to stand in firm, unwavering peaks on the beater when it is lifted out of the bowl.
To assemble, slice each cake in half horizontally to make four thin layers.
Place one of the layers, cut side up, on a serving plate and, with a metal spatula, spread it with about 1/2 cup of the pureed strawberries.
Repeat two more times, spreading the third layer of the cake with all of the remaining pureed berries.
Carefully put the fourth layer on the top, cut side down.
Then spread the whipped cream smoothly over the top and sides of the strawberry cake.
Cut a dozen or so of the reserved 2 cups of whole strawberries in half lengthwise and set them on the plate, cut side up, in a ring around the cake.
Arrange the remaining whole strawberries attractively on the top of the cake and serve at once.